The centre of the circle $r^2-4r(\cos \theta+\sin \theta)-4=0$ in Cartesian coordinates is

$A$ circle $x^2 + y^2 + 2gx + 2fy + c = 0$ passing through $(4, -2)$ is concentric to the circle $x^2 + y^2 - 2x + 4y + 20 = 0$. Then the value of $c$ is:

The centre and radius of the circle $2x^2 + 2y^2 - x = 0$ are

For what value of $t$ do the four distinct points $(2, 3), (0, 2), (4, 5),$ and $(0, t)$ lie on a circle?

If a circle passing through $(1, -2)$ has $x - y = 2$ and $2x + 3y = 14$ as its diameters,then the radius of the circle is

If a circle passing through the point $(-1, 0)$ touches the $y-$ axis at $(0, 2)$,then the length of the chord of the circle along the $x-$ axis is